Errors on JavaScript product operations

Open the website inspector in Chrome (cmd + alt + i or ctrl + shift + i) and try the following:

javascript
1.85 * 3 // 5.550000000000001

You are getting an error because JavaScript uses double-precision numbers. These numbers occupy 64 bits:

terminal
S  EEEEEEEEEEE  FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
0  1         11 12                                                 64

• The first bit represents the sign of the number (S).
• The next eleven bits are the exponent bits (E).
• The final 52 are the fraction (F).

The value that you see after multiplying is calculated by the double-precision numbers formula:

$v = (-1)^s · (1 + f) · 2^{(e - b)}$

Where s is the sign, f is the fraction, e is the exponent and b is the bias.

Example converting from IEEE 754 Form

Going back to the previous example. Let's suppose that you want to generate the following decimal number:

javascript
let v = 5.55

The closest binary number that generates the previous result would be:

terminal
0  10000000001  0110001100110011001100110011001100110011001100110100
0  1         11 12                                                 64

Let's demonstrate it by applying the previous formula and some JavaScript:

javascript
let S = '0'
let E = '10000000001'
let F = '0110001100110011001100110011001100110011001100110100'

// Normal conversion from binary to base 10
let s = parseInt(S, 2)

// Normal conversion from binary to base 10
let e = parseInt(E, 2)

// This is how to calculate the fraction of double precision numbers
// The following line is equivalent to 0 · 2⁻¹ + 1 · 2⁻² + 1 · 2⁻³ + 0 · 2⁻⁴ + ... + 0 · 2⁻⁵²
let f = [...F].reduce((acc, num, i) => acc += num * 2 ** (-(i + 1)), 0)

// Let's define the bias (127 for 32 bits and 1023 for 64 bits)
let b = 1023

// Applying the formula
let v = (-1) ** s * (1 + f) * (2 ** (e - b)) 

// This is the final result
//  5.550000000000001